Search

The contribution of this paper is the proposal of a new receding horizon trajectory generation method for stratospheric airships' return phase. Since the energy consumption, wind field and path constraints are restrictions during the return phase of airships at low altitude, it is crucial to develop novel trajectory optimization methods to ensure that the airship returns to the landing site. In this article, optimization objects and conditions of the return trajectory were constructed, considering the energy consumption and wind field. Then, a modified interior point method (MIP) was used to transform the inequality constraint, which is a simple and adaptable method used to improve the solving efficiency on the basis of a modified multiple shooting method (MMS). In addition, an adaptive gradient descent regulator was improved to reduce the influence on the optimization result due to different selections of the initial search point, and the convergence was made faster and more stable. Finally, considering the performance and path constraints of the airship, the effectiveness of the scheme was verified by numerical experiments under wind effects and a comparison of different methods. [ABSTRACT FROM AUTHOR]

The contribution of this paper is the proposal of a new receding horizon trajectory generation method for stratospheric airships’ return phase. Since the energy consumption, wind field and path constraints are restrictions during the return phase of airships at low altitude, it is crucial to develop novel trajectory optimization methods to ensure that the airship returns to the landing site. In this article, optimization objects and conditions of the return trajectory were constructed, considering the energy consumption and wind field. Then, a modified interior point method (MIP) was used to transform the inequality constraint, which is a simple and adaptable method used to improve the solving efficiency on the basis of a modified multiple shooting method (MMS). In addition, an adaptive gradient descent regulator was improved to reduce the influence on the optimization result due to different selections of the initial search point, and the convergence was made faster and more stable. Finally, considering the performance and path constraints of the airship, the effectiveness of the scheme was verified by numerical experiments under wind effects and a comparison of different methods.

This paper presents an iterative convex programming algorithm for the complex ascent trajectory planning problem. Due to the nonlinear dynamics and constraints, ascent trajectory planning problems are always difficult to be solved rapidly. With deterministic convergence, convex programming is becoming increasingly attractive to such problems. In this paper, first, path constraints (dynamic pressure, load and bending moment) are convexified by a change of variables and a reasonable approximation. Then, based on the Newton–Kantorovich/Pseudospectral (N–K/PS) approach, the dynamic equations are transcribed into linearized algebraic equality constraints with a given initial guess, and the ascent trajectory planning problem is formulated as a convex programming problem. At last, by iteratively solving the convex programming problem with readily available convex optimization methods and successively updating the initial guess with the Newton–Kantorovich iteration, the trajectory planning problem can be solved accurately and rapidly. The convergence of the proposed iterative convex programming method is proved theoretically, and numerical simulations show that the method proposed can potentially be implemented onboard a launch vehicle for real-time applications. [ABSTRACT FROM AUTHOR]

Without necessarily assuming that switching sequences are fixed, a dynamic optimization method is proposed for optimal control of state-dependent switched systems. First, a parameterization method is developed to parameterize the switching instants and control vectors to facilitate the calculation of the gradient information, and then the original problem becomes a finite-dimensional mixed discrete–continuous nonlinear program as the switching sequence is discrete and the other variables are continuous. Secondly, the mixed discrete–continuous nonlinear program is transformed into an equivalent problem that contains only continuous variables by relaxing 0 and 1 discrete variables into continuous variables between 0 and 1 and adding proper linear and quadratic constraints. Thirdly, the formulas to compute the gradients of the objective function with respect to all the arguments are derived by solving the variational systems and a two-point boundary value differential algebraic equations (DAEs). Fourthly, an algorithm is proposed to locate a feasible point satisfying the Karush–Kuhn–Tucker (KKT) conditions to a specified tolerance of dynamic optimization of switched systems (DOSS) while guaranteeing feasibility of inequality path constraints, and the finite convergence of the algorithm is proved. Finally, the performance of the algorithm is analyzed via a numerical example. [ABSTRACT FROM AUTHOR]

An algorithm is proposed for locating a feasible point satisfying the KKT conditions to a specified tolerance of feasible inequality-path-constrained dynamic programs (PCDP) within a finite number of iterations. The algorithm is based on iteratively approximating the PCDP by restricting the right-hand side of the path constraints and enforcing the path constraints at finitely many time points. The main contribution of this article is an adaptation of the semi-infinite program (SIP) algorithm proposed in Mitsos (2011) to PCDP. It is proved that the algorithm terminates finitely with a guaranteed feasible point which satisfies the first-order KKT conditions of the PCDP to a specified tolerance. The main assumptions are: (i) availability of a nonlinear program (NLP) local solver that generates a KKT point of the constructed approximation to PCDP at each iteration if this problem is indeed feasible; (ii) existence of a Slater point of the PCDP that also satisfies the first-order KKT conditions of the PCDP to a specified tolerance; (iii) all KKT multipliers are nonnegative and uniformly bounded with respect to all iterations. The performance of the algorithm is analyzed through two numerical case studies. [ABSTRACT FROM AUTHOR]

Controller tuning is needed to select the optimum response for the controlled process. This work presents a new tuning procedure of PID controllers with safety and response quality measures on a non-linear process model by optimization procedure, with a demonstration of two tanks in series. The model was developed to include safety constraints in the form of path constraints. The model was then solved with a new optimization solver, NLPOPTI, which uses a prima!-dual interior point method with a novel non-monotone line search procedure with discretized penalty parameters. This procedure generated a grid of optimal PID tuning parameters for various switching of steady-states to be used as a predictor of PID tunings for arbitrary transitions. The interpolation of tuning parameters between the available parameters was found to be capable to produce state profiles with no violation on the safety measures, while maintaining the quality of the solution with the final set points targeted achievable. [ABSTRACT FROM AUTHOR]

Numerically discretized dynamic optimization problems having active inequality and equality path constraints that along with the dynamics induce locally high index differential algebraic equations often cause the optimizer to fail in convergence or to produce degraded control solutions. In many applications, regularization of the numerically discretized problem in direct transcription schemes by perturbing the high index path constraints helps the optimizer to converge to useful control solutions. For complex engineering problems with many constraints it is often difficult to find effective nondegenerate perturbations that produce useful solutions in some neighborhood of the correct solution. In this paper we describe a numerical discretization that regularizes the numerically consistent discretized dynamics and does not perturb the path constraints. For all values of the regularization parameter the discretization remains numerically consistent with the dynamics and the path constraints specified in the original problem. The regularization is quantifiable in terms of time step size in the mesh and the regularization parameter. For fully regularized systems the scheme converges linearly in time step size. The method is illustrated with examples. [ABSTRACT FROM AUTHOR]

We present an extension of the direct multiple shooting method for the solution of optimal control problems with path constraints. The extension allows for fully automatic and accurate treatment of path constraints in case of relatively coarse discretizations of the control while maintaining a prescribed resolution of the system dynamics. It is based on a reduction principle from classical theory of semi-infinite programming. On the basis of these theoretical foundations the minima tracking algorithm is designed. We theoretically and numerically compare this method with a sampling technique. The numerical test case is a well-known, difficult space shuttle reentry benchmark problem. [ABSTRACT FROM AUTHOR]

Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-defining and the introduction of new classes of valid inequalities. [ABSTRACT FROM AUTHOR]

We address some advantages of nonlinear programming (NLP)-based methods for inequality path-constrained optimal control problems. The analysis is carried out on the differential algebraic equation (DAE)-constrained optimization problem presented in [J. T. Betts and S. L. Campbell, Discretize Then Optimize, Technical report M&CT-TECH-03-01, The Boeing Company, Chicago, IL, 2003], which relates to a one-dimensional transient heat conduction problem. This optimal control problem possesses the essential characteristics of a typical inequality pathconstrained optimal control problem. In a DAE-constrained optimization setting, the index of the path constraint can be arbitrarily increased, and this facilitates the study of the effect of the index of a path constraint on the solution obtained by using an NLP-based methodology. We show that the direct transcription approach leads to a problem whose limiting behavior does not satisfy the linear independence constraint qualification but satisfies the weaker Mangasarian-Fromowitz constraint qualification. The direct transcription approach leads to a convex quadratic programming problem with the result that the control obtained is the unique global minimizer. We also contrast this with the classical indirect approach where we show that it is difficult to numerically integrate the ODEs over the constrained arc because of stability and error control reasons. These observations explain some of the results of Betts and Campbell. This supports the fact that NLP-based methodologies have additional flexibility with respect to constraint qualifications, and this can be put to use in the case of inequality path-constrained optimal control problems to obtain well-defined solutions. [ABSTRACT FROM AUTHOR]

Several classes of path constraints for semistructured data are analysed and a number of decidability and complexity results proved for such constraints. While some of these decidability results were known before, it is believed that the improved complexity bounds are new. Proofs are based on techniques from modal logic and automata theory. This modal logic perspective sheds additional light on the reasons for previously known decidability and complexity results. [ABSTRACT FROM PUBLISHER]

In this paper an algorithm for extracting a stable differential-algebraic subsystem from a path-constrained dynamical system is proposed. The subsystem may be integrated directly by a differential-algebraic system integrator to evaluate constraints in shooting- or multiple shooting- type direct methods for solving path-constrained dynamic optimization problems. The algorithm appends algebraic constraints to the unconstrained ordinary differential equation subsystem based on a stability estimate for the resulting differential-algebraic system. The logarithmic norm is used to compute a stability estimate for index 1 and index 2 subsystems. The working of the algorithm is illustrated with examples. [ABSTRACT FROM AUTHOR]

An algorithm is proposed for locating an optimal solution satisfying the KKT conditions to a specified tolerance to inequality-path-constrained dynamic optimization (PCDO) problem within finite iterations. This algorithm solves the optimization problem by iteratively approximating the original optimization problem through adaptive convexification of its lower level programs. In the process of convexification of the lower level programs, α BB technique is used adaptively to construct convex relaxations of a path constraint in each time subinterval. Compared to the result in (Fu, Faust, Chachuat, & Mitsos, 2015), the distinguishing feature is that the proposed algorithm avoids numerically solving the non-convex lower level program to global optimality at each iteration. Two numerical examples are shown to demonstrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

Optimal control problems with inequality path constraints (IPCs) are present in several engineering problems described by partial differential equations (PDE). We propose an algorithm to solve PDE-constrained dynamic optimization problems with guaranteed satisfaction of IPCs. The algorithm is based on a solution of a sequence of approximated dynamic optimization problems following the strategy of Fu et al. (Automatica, 2015). For the approximation, the path constraint is imposed only on a set of discrete points and is thus relaxed. At the same time, it is restricted by an adaptive back-off and by the error bound of the PDE solution. The approximation is iteratively improved: at iterations without violation, the restriction is reduced; at iterations with violation, the point of maximal violation is added to the discrete set. Under mild assumptions, we prove finite termination of the algorithm. We test the algorithm on an optimal grade change of a tubular reactor. [ABSTRACT FROM AUTHOR]